Giovanni Lagnese wrote:
> But one can take as primitive also the concept of powerset. So there
> must be a philosophical reason for taking as primitive the concept of
> natural numbers but not the concept of powerset. And this reason can
> not be the predicativism.
No, you cannot take the concept of power set as primitive if you
don't even know what sets are. This aspect of my last message
seems not to have been understood.
The impatience with predicativism seen here seems to arise from an
implicit acceptance of a platonic conception of set theory. It is
taken for granted that there are these abstract entities, sets, which
reside in some metaphysical world, and starting from this point one
wants to know why predicativists accept some parts of that world and
not others. I do not accept any part of this picture and I think that
anyone who does has a lot of work to do in defending his own views
before going after rival stances like predicativism. Quoting Hartley
Slater ("Grammar and sets", to appear in the Australasian Journal of
Philosophy): "the very notion of a set ... is based on a series of
grammatical confusions." (Also: "The expectation ... has been that
one must look elsewhere for _another object_ to be the pair of apples.
But this supposed other object is a grammatical mirage.")
I could turn the point around and ask why, if you are willing to
accept the notion of power set as primitive (for all x, there is
a set P(x) such that y is in P(x) <=> y is a subset of x), you do
not more generally accept full comprehension (for all x and any
formula p, there is a set z such that y is in z <=> p(x,y) holds).
Can you give a principled explanation of why in your supposed
platonic universe of sets one holds and the other doesn't?
>From my perspective, we cannot accept any set as primitive. In
order to meaningfully use the language of set theory one must
explicitly identify an interpretation, i.e., we must specify
which objects are to play the role of sets and for which pairs
of these objects the membership relation is supposed to obtain.
On this view it is obvious that we cannot simply posit the
existence of power sets. We have to show how they can be built.
And there is a fundamental circularity in the concept of power
set which obstructs us from doing this in general. See the
bottom of page 10 of my paper "Mathematical conceptualism".
The argument that identifying a structure which is to play the
role of the natural numbers is somehow circular puzzles me.
I make a mark on a piece of paper, then I make another mark
next to it, and so on. To say that this doesn't work because
I can't actually physically make infinitely many marks seems
to me to take physics as being more fundamental than mathematics,
a view which has been explicitly advocated in recent posts in
another thread, but which I cannot accept. Mathematics, i.e., the
realm of logical possibility, is more primitive than the realm of
physical possibility. So I accept (a structure which plays the
role of) the natural numbers because I can see clearly how its
construction could be carried out, which is enough. I do not
see how a structure playing the role of the power set of the
natural numbers could be constructed in any comparable way.
In my paper "Analysis in J_2" I outline how the vast bulk of
mainstream mathematics can be carried out in an explicitly
constructed predicative universe (and essentially the same
point has been made by many other authors; see the references
in the J_2 paper). The J_2 structure in fact provides a much
better fit with normal mathematical practice than the fictional
Cantorian universe with its vast cardinals which play absolutely
no role in mainstream mathematics and its set-theoretic pathologies
which only distract and irritate the core mathematician.
Predicativists need not consider themselves on the defensive.
It is only a matter of time before it will be widely recognized
that our approach both avoids the metaphysical nonsense of Cantorian
set theory and leads to a conception of the mathematical universe
which is in greater harmony with normal mathematics.
Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu